To solve the problem step by step, we need to calculate the total distance traveled by the motorcyclist during each phase of the journey and then find the average speed.
### Step 1: Calculate the distance during acceleration
The motorcyclist accelerates from rest with an acceleration of \(2 \, \text{m/s}^2\) for \(10 \, \text{s}\).
Using the formula for distance under uniform acceleration:
\[
S_1 = ut + \frac{1}{2} a t^2
\]
where:
- \(u = 0 \, \text{m/s}\) (initial velocity),
- \(a = 2 \, \text{m/s}^2\) (acceleration),
- \(t = 10 \, \text{s}\) (time).
Substituting the values:
\[
S_1 = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 = 0 + \frac{1}{2} \cdot 2 \cdot 100 = 100 \, \text{m}
\]
### Step 2: Calculate the constant velocity
After accelerating, the motorcyclist moves at a constant velocity for \(20 \, \text{s}\). To find this velocity, we use:
\[
v = u + at
\]
At the end of the acceleration phase:
\[
v = 0 + 2 \cdot 10 = 20 \, \text{m/s}
\]
### Step 3: Calculate the distance during constant velocity
Now, the distance traveled during the constant velocity phase is:
\[
S_2 = v \cdot t = 20 \cdot 20 = 400 \, \text{m}
\]
### Step 4: Calculate the distance during deceleration
Finally, the motorcyclist decelerates at \(1 \, \text{m/s}^2\) until coming to rest. We need to find the time taken to stop.
Using the equation:
\[
v = u + at
\]
Setting \(v = 0\) (final velocity) and \(u = 20 \, \text{m/s}\):
\[
0 = 20 - 1 \cdot t \implies t = 20 \, \text{s}
\]
Now, we can calculate the distance during deceleration using:
\[
S_3 = ut + \frac{1}{2} a t^2
\]
where \(u = 20 \, \text{m/s}\), \(a = -1 \, \text{m/s}^2\), and \(t = 20 \, \text{s}\):
\[
S_3 = 20 \cdot 20 + \frac{1}{2} \cdot (-1) \cdot (20)^2 = 400 - 200 = 200 \, \text{m}
\]
### Step 5: Calculate total distance and total time
Now, we can find the total distance traveled:
\[
S_{\text{total}} = S_1 + S_2 + S_3 = 100 + 400 + 200 = 700 \, \text{m}
\]
The total time taken is:
\[
t_{\text{total}} = 10 + 20 + 20 = 50 \, \text{s}
\]
### Step 6: Calculate average speed
Finally, the average speed is given by:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{700 \, \text{m}}{50 \, \text{s}} = 14 \, \text{m/s}
\]
### Final Answer
The average speed for the complete journey is \(14 \, \text{m/s}\).
